Average Error: 0.3 → 0.3
Time: 7.4s
Precision: 64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
↓
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)} \cdot {\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)}\right)\]
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
↓
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)} \cdot {\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)}\right)
double code(double x, double y, double z, double t) {
	return ((((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0)));
}
↓
double code(double x, double y, double z, double t) {
	return ((((x * 0.5) - y) * sqrt((z * 2.0))) * (pow(sqrt(exp(t)), (t / 2.0)) * pow(sqrt(exp(t)), (t / 2.0))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Target

Original0.3
Target0.3
Herbie0.3
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)}\]

Derivation

  1. Initial program 0.3

    \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{\color{blue}{1 \cdot 2}}}\]

  4. Applied times-frac0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t}{1} \cdot \frac{t}{2}}}\]

  5. Applied exp-prod0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{\frac{t}{1}}\right)}^{\left(\frac{t}{2}\right)}}\]

  6. Simplified0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\color{blue}{\left(e^{t}\right)}}^{\left(\frac{t}{2}\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\color{blue}{\left(\sqrt{e^{t}} \cdot \sqrt{e^{t}}\right)}}^{\left(\frac{t}{2}\right)}\]

  9. Applied unpow-prod-down0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)} \cdot {\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)}\right)}\]

  10. Final simplification0.3

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)} \cdot {\left(\sqrt{e^{t}}\right)}^{\left(\frac{t}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2))) (exp (/ (* t t) 2))))